Troops, Singletons -- and Stuff

J:

"A committe implies a group of people".

Or as I would say,

"implicates"

From wiki:

"a singleton is a set with exactly one element. For example, the set {0} is a singleton."

"I found a set in the room".
"A set of what?"
" {0} -- you said my shoes would be there but they aren't"

---

Wiki:


"Note that a set such as

{{1, 2, 3}}

is also a singleton."

-----

A: You have THREE of them?
B: Yes. But I have arranged them in such a way -- {{1, 2, 3}} -- that they, strictly, only count as one.
A: Gebirge!


Wiki:

"the only element is a set (which itself is however not a singleton). A set is a singleton iff its cardinality is 1."

---- So, if the cardinality of the 'berge' is 1 (The Himalaya) a mountain can be its own range. Or not.

Wiki:

"In the set-theoretic construction of the natural numbers, the number 1 is defined as the singleton {0}. In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing."

Pretty much as elsewhere. In Biology, too, the existence of plurality is a consequence of (the axiom of) pairing (or 'mating', in the vernacular).

Wiki:

"for any set A that axiom applied to A and A asserts the existence of {A,A}, which is the same as the singleton {A} (since it contains A, and no other set, as element).
If A is any set and S is any singleton, there exists precisely one function from A to S, the function sending every element of A to the one element of S. In topology, a space is a T1 space if and only if every singleton is closed. Structures built on singletons often serve as terminal objects or zero objects of various categories."


---- such as one-mountain mountain ranges -- topologically.

Wiki:

"The statement above shows that the singleton sets are precisely the terminal objects in the category Set of sets. No other sets are terminal. Any singleton can be turned into a topological space in just one way (all subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and continuous functions. No other spaces are terminal in that category."

--- "ontological pluralism" (needless):

"Any singleton can be turned into a group."

--- "My classes are usually big. His name is Jim."
--- "You are a lucky one. My class is not so big, but the uni still pays for it. Oxford you know."

Wiki:

"Any singleton can be turned into a group in just one way (the unique element serving as identity element). These singleton groups are zero objects in the category of groups and group homomorphisms."

--- "That's strictly, not a one-mountain mountain range. It is a group homomorphism, if you must."

---

Wiki:

"No other groups are terminal in that category. Let S be a class defined by a Boolean-valued function . Then:

"S is called a singleton iff b is equal to some function c, with c(x) = (x = y) for some c." (or something) (check with wiki for formalism -- it's not plain text, by far!).

"Traditionally, this definition was introduced by Whitehead and Russell[1] along with the definition of the natural number 1, a as b , where c." Very complex formulae here, in hypertext. What's wrong with wiki?

See also
Class (set theory)

References
1.^ Whitehead, Alfred North; Bertrand Russell (1861). Principia Mathematica. pp. 37.